Concavity of the Auxiliary Function for Classical-Quantum Channels
Journal
IEEE Transactions on Information Theory
Journal Volume
62
Journal Issue
10
Start Page
5960
End Page
5965
ISSN
00189448
Date Issued
2016-10
Author(s)
Min-Hsiu Hsieh
Abstract
The auxiliary function of a classical channel appears in two fundamental quantities, the random coding exponent and the sphere-packing exponent, which yield upper and lower bounds on the error probability of decoding, respectively. A crucial property of the auxiliary function is its concavity, and this property consequently leads to several important results in finite blocklength analysis. In this paper, we prove that the auxiliary function of a classical-quantum channel also enjoys the same concavity property, extending an earlier partial result to its full generality. We also prove that the auxiliary function satisfies the data-processing inequality, among various other important properties. Furthermore, we show that the concavity property of the auxiliary function enables a geometric interpretation of the random coding exponent and the sphere-packing exponent of a classical-quantum channel. The key component in our proof is an important result from the theory of matrix geometric means.
Type
journal article